ASYMMETRIC AND SINGLY SYMMETRIC SECTIONS IN COMPRESSION

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TECHNICAL
42
Steel Construction Vol. 32 No. 1 February 2008
Last year the SAISC took its “Design of light industrial buildings” course to a
number of cities and towns throughout South Africa. One of the things that
became apparent to me while lecturing on the course was that a number of
design engineers do not fully appreciate or understand the requirements of
clause 13.3.2 of the current South African hot-rolled steel design code.
Consequently, they are in the dark with regards to the design of asymmetric and
singly symmetric sections. This article, dealing with the basics of concentric
elastic buckling of columns, is the first in a series to address the design of these
types of sections.
My guess is that most structural engineers (myself included) grew up on a diet of
elastic flexural buckling. Euler’s celebrated formula was drummed into our heads
from early on in our academic careers. At an undergraduate level, torsional and
flexural-torsional buckling were topics that were possibly not covered in much
depth or were topics that were way too strenuous to contemplate with heavy eye-
lids. When the first limit states design code for structural steelwork
(SABS0162-1:1993) was introduced in South Africa, it lacked guidance on how to
design singly symmetric or asymmetric compression members. To address this state
of affairs, the SAISC published a technical note in Steel Construction
(November/December 1993) which contained a proposed methodology that has
now essentially been incorporated into the current design code as clause 13.2.2.
Those engineers who are familiar with the South African limit states design code
for cold-formed steelwork would have recognised the singly symmetric elastic
buckling equations since they are similar to those given in SANS10162-2:1993.
Other than nomenclature, the main difference is that the cold-formed code implic-
itly considers the section to be symmetrical about the x axis (as opposed to the
y-axis in 10162-1). The cubic equation for elastic buckling of asymmetric sections
was, however, unfamiliar territory for many engineers.
Understanding the nature of elastic buckling of columns, be it flexural, torsional
or flexural-torsional, can be enhanced by unpacking the equation given in
clause 13.3.2 c) viz:
The derivation of this equation can be found in many classic mechanics text-
book. So let’s not get into that level of detail here except to point out that it is
generally expressed in terms of elastic buckling loads and not stresses.
The solution to this equation gives the elastic buckling stress, f
e
, of a column.
The terms f
ex
, f
ey
and f
ez
refer to the elastic buckling stress of the column about
it’s strong axis, about it’s weak axis and the elastic torsional buckling stress
respectively. The polar moment of inertia of the centroid about the shear centre
is denoted by r
o
. The terms x
o
and y
o
are defined as the principal coordinates of
the shear centre with respect to the centroid of the cross-section. To clarify
what this means, consider diagram 1 (shown left) of a general open cross-
section. The axes labelled X and Y are the principal centroidal axes of the
section, point C identifies the location of the centroid and point O the shear
centre. x
o
and y
o
are the dimensions as shown.
It is a well know fact that if the cross-section of a column has two axes of
symmetry, then the shear centre coincides with the centroid of the cross-
Equation 1.
Diagram 1.
ASYMMETRIC
AND SINGLY
SYMMETRIC
SECTIONS IN
COMPRESSION
By David Blitenthall, development
engineer, SAISC
My guess is that most structural
engineers (myself included) grew up
on a diet of elastic flexural buckling.
Euler’s celebrated formula was
drummed into our heads from early
on in our academic careers. At an
undergraduate level, torsional and
flexural-torsional buckling were
topics that were possibly not covered
in much depth or were topics that
were way too strenuous to
contemplate with heavy eye-lids.
TECHNICAL
Steel Construction Vol. 32 No. 1 February 2008
43
Equation 3.
Equation 2.
section. In this case x
o
= y
o
= 0 mm. Substituting these values into equation 1,
gives the following:
The three solutions to this equation are obviously f
e
= f
ex
or f
ey
or f
ez
. The
column will therefore buckle at the lowest of these stresses and in a correspon-
ding mode viz. buckling either occurs by flexure about the strong axis, flexure
about the weak axis or by torsion. It is also important to note that these three
modes are independent in this case.
For any section having only one axis of symmetry, the shear centre will be on
that axis but generally not at the centroid. In the case where the y axis is the
axis of symmetry (a T-section for example), x
o
= 0 mm. Substituting this into
equation (1), give the following:
Substituting Ω = 1 – (y
o
/ r
o
)
2
and solving the term in square parenthesis using
the quadratic formula, the solutions to this equation are f
e
= f
ex
and
.
Such a column will therefore buckle at the lowest of the three stresses,
either in a flexural mode about the x-axis at f
ex
or in a torsional-flexural
mode at the lowest root given by the quadratic equation – this root being
lower than either f
ey
or f
ez
. In this case, flexural buckling about the x-x axis
is independent while flexural buckling about the y-y axis and torsional buck-
ling are coupled. In the case where the x axis is the axis of symmetry (a
channel section for example), y
o
= 0mm and the solution to equation (1) can
be found by interchanging the y and x terms such that f
e
= f
ey
and
.
For an asymmetric section, x
o
Ω0 and y
o
Ω0 and the solution of f
e
requires
finding the lowest root of equation (1). The buckling modes in this instance
are interdependent and the lowest buckling mode is always less than either
f
ex
, f
ey
or f
ez
.
The next article on this subject will take a look at how this basic elastic buckling
theory is applied in SANS10162-1.